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Possible error in Bayesian bootstrap

After my last post on Bayesian bootstrap I got a question why the sample from Dirichlet distribution is taken as weights for calculating mean in the procedure and not as weights used for sampling from the original data set. Actually this mistake is subtle and occurs even in textbooks, see example Chernick (2008), page 122. In this post I want to clarify the issue.In the example I give correct bootstrap and Bayesian bootstrap procedures and wrong ones. The wrong Bayesian bootstrap follows description from Chernick (2008), page 122 (that is equivalent to the comment to my last post).

We can see that wrong Bayesian bootstrap has an equivalent in standard bootstrap approach that is generated by repeating the sampling twice (sampling from a sample) and it clearly increases dispersion of the results.